INFINITELY MANY SOLUTIONS FOR A NONLOCAL PROBLEM

Abstract

Consider a class of nonlocal problems $ \left\{\begin{array}{lr} -\left(a-b \int_{\Omega}|\nabla u|^{2} d x\right) \Delta u=f(x, u), & x \in \Omega, \\ u=0, & x \in \partial \Omega, \end{array}\right.$ where $ a>0, b>0 $, $ \Omega\subset \mathbb{R}^N $ is a bounded open domain, $ f:\overline{\Omega} \times \mathbb R \longrightarrow \mathbb R $ is a Carath$ \acute{\mbox{e}} $odory function. Under suitable conditions, the equivariant link theorem without the $ (P.S.) $ condition due to Willem is applied to prove that the above problem has infinitely many solutions, whose energy increasingly tends to $ a^2/(4b) $, and they are neither large nor small.